# Understanding an integral rule.

1. Jul 22, 2015

### christian0710

Hi I'm tying to understand notations in Integration, and would really appreciate some help making sure that my understanding is right.

My books writes

Let u and v be functions of x whose domains are an open interval I, and suppose du and dv exist for every x in I.

Then it defines

1) ∫(du)= u + C
2) ∫(c*du) = c*∫(du)

and
3) ∫(cos(u)du = sin u +C

Now i do understand the first 2, but I want to make sure i understand the 3rd rule.

If u is a function of x with the equation u(x)=x^2

Then the derivative
du/dx= 2x

The differential
du=u'(x)dx

Now if it's true that du=u'(x)dx
Then it does make sense that ∫du =u+C because ∫du=∫u'(x)*dx and the integral of the derivative if u is u.

But if u=x^2

Then ∫(cos(x^2)*du = ∫(cos(x^2)*(2x)dx and this is = sin(x^2) + C as the statement above says.
because the derivative of sin(x^2) = cos(x^2)*(2x).

Is this the right interpretation?

Last edited: Jul 22, 2015
2. Jul 22, 2015

### cpsinkule

why is ∫(cos(x^2)*(2x)dx not equal to sin(x^2) + C? you said in the line below it that
which is exactly what appears under the integral sign.

3. Jul 22, 2015

### christian0710

Oh forgive me, that NOT was a BIG mistake ;)
You are right, it is e equal to it.